`ai + 3j + 4k and sqrt(b) i+5k` are two vectors, where `a, b gt 0` are two vectors, where `a, b ge 0` are two scalars, then the length of the vectors is equal for

Options

A. all value of (a, b)
B. only finite number of values of `(a, b)`
C. infinite number of values of (a, b)
D. no value of (a, b).

Answer

To solve the problem, we need to find the conditions under which the magnitudes of the two vectors are equal. The vectors given are:

1. \( \mathbf{p} = a \mathbf{i} + 3 \mathbf{j} + 4 \mathbf{k} \)
2. \( \mathbf{q} = \sqrt{b} \mathbf{i} + 0 \mathbf{j} + 5 \mathbf{k} \)

### Step 1: Calculate the magnitude of vector \( \mathbf{p} \)

The magnitude of a vector \( \mathbf{v} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) is given by:

\[ |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \]

For vector \( \mathbf{p} \):

\[ |\mathbf{p}| = \sqrt{a^2 + 3^2 + 4^2} \]
\[ |\mathbf{p}| = \sqrt{a^2 + 9 + 16} = \sqrt{a^2 + 25} \]

### Step 2: Calculate the magnitude of vector \( \mathbf{q} \)

For vector \( \mathbf{q} \):

\[ |\mathbf{q}| = \sqrt{(\sqrt{b})^2 + 0^2 + 5^2} \]
\[ |\mathbf{q}| = \sqrt{b + 25} \]

### Step 3: Set the magnitudes equal to each other

To find the condition for which the lengths of the vectors are equal, we set their magnitudes equal:

\[ \sqrt{a^2 + 25} = \sqrt{b + 25} \]

### Step 4: Square both sides to eliminate the square roots

Squaring both sides gives:

\[ a^2 + 25 = b + 25 \]

### Step 5: Simplify the equation

Subtracting 25 from both sides results in:

\[ a^2 = b \]

### Step 6: Find the relationship between \( a \) and \( b \)

From the equation \( a^2 = b \), we can express \( a \) in terms of \( b \):

\[ a = \sqrt{b} \]

### Conclusion

Since \( a \) and \( b \) are scalars and must be greater than or equal to 0, the relationship \( a = \sqrt{b} \) holds for all \( b \geq 0 \). Therefore, there are infinitely many pairs \( (a, b) \) that satisfy this condition.